Since the destruction of the Temples, Jews have struggled with the enduring meaning and legacy of the sacrificial rites.

Animal sacrifices are described in detail in the book of Leviticus and were offered throughout the period of the First and Second Temples.

That Gentiles as well as Jews brought sacrifices to the Temple is implied in the prayer of Solomon when he dedicated the Temple (I Kings 8:41-3) and in the declaration by the prophet that the Temple will be a house of prayer for all peoples (Isaiah 56:7).
The Rabbis say (Hullin 13b): 'Sacrifices are to be accepted from Gentiles as they are from Jews,' although this saying dates from after the destruction of the Temple.

The significance of the role of the sacrifice in the Temple period is expressed in the saying in Ethics of the Fathers (1.3) that the world stands on three things, the Torah, the service in the Temple, and benevolence. That occasionally the prophets seem to decry the offering of sacrifices (e.g. Amos 5:21-4; Isaiah 1:11-13) is explained in the Jewish tradition, and this might well be the case, that the prophets only object to sacrifices used as an attempt to buy off God while practicing iniquities.

The ancients did not have the scruples of many moderns about offering up 'poor defenseless animals.' People did and still do kill animals for food and, apart from the wholly consumed burnt-offering, the meat of all the other sacrifices was eaten either by the priests or by those who brought the sacrifices.

Order Kodashim

The whole of the Order Kodashim in the Mishnah is devoted almost entirely to the details of how the sacrifices were to be offered. This order was compiled in its present form after the destruction of the Temple but a good deal of the material undoubtedly stems from traditions in Temple times.

The order was studied, even though the laws of sacrifices had fallen into abeyance, in the belief that these were all part of the divinely revealed Torah and that the sacrificial system would one day be restored.

Medieval Explanations

Various explanations have been advanced in medieval and modern times for why God commanded that sacrifices be offered to Him. This kind of thinking was unknown to the Talmudic Rabbis. For them it was enough that God had ordained that sacrifices should be offered and they saw no need to ask why. But from the Middle Ages onwards, attempts were made to provide what seemed to be to those who made them plausible reasons for the sacrificial cult.

According to Maimonides, in his Guide of the Perplexed, the sacrifices were ordained in order to wean the people of Israel away from idolatry, as if God were to say: if the idea of offering sacrifices has taken too strong a hold on you to be totally eradicated, at least offer the sacrifices in a central place and observe the rules in order to avoid the excesses practiced by the idolaters when they sacrifice to their gods.